Summary: We consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies \(W:\mathrm{GL}^+ (n)\rightarrow \mathbb{R}\) with an additive volumetric-isochoric split, i.e. \[ W(F)=W_{\mathrm{iso}}(F)+W_{\mathrm{vol}}(\det F)= \widetilde{W}_{\mathrm{iso}} \left(\frac{F}{\sqrt{\det F}}\right) +W_{\mathrm{vol}} (\det F)\,, \] which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of “least” rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function \[ W_{\mathrm{magic}}^+ (F)=\frac{\lambda_{\max}}{\lambda_{\min}} -\log \frac{\lambda_{\max}}{\lambda_{\min}}+\log \det\, F=\frac{\lambda_{\max}}{\lambda_{\min}}+2\log \lambda_{\min} \] is quasiconvex. In addition, we demonstrate that under affine boundary conditions, \(W_{\mathrm{magic}}^+ (F)\) allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of D. L. Burkholder [Astérisque 157–158, 75–94 (1988; Zbl 0656.60055)] and T. Iwaniec [Banach Cent. Publ. 48, 119–140 (1999; Zbl 0942.46016)] in the field of complex analysis.